Introduction

Binary logistic regression handles two-category outcomes. But many epidemiologic outcomes have more than two categories:

General health: Excellent / Very Good / Good / Fair / Poor
Disease stage: I / II / III / IV
Treatment choice: Drug A / Drug B / Drug C / No treatment
BMI category: Underweight / Normal / Overweight / Obese

When the response variable has \(K > 2\) categories, we use polytomous logistic regression (also called multinomial logistic regression). When the categories have a natural order, we can take advantage of that ordering with ordinal logistic regression.

Today’s roadmap (Kleinbaum Ch. 23): 1. Polytomous (multinomial) logistic regression for nominal outcomes 2. Ordinal logistic regression for ordered outcomes 3. The proportional odds assumption 4. Choosing between models

Why Not Just Run Multiple Binary Models?

Suppose general health has 3 categories: Excellent, Good, Poor. A naive approach would be to dichotomize and run a binary logistic regression repeatedly:

Excellent vs. not
Good vs. not
Poor vs. not

Problems with this approach:

Information loss. Dichotomizing throws away the distinction between the other categories.
Inconsistent samples. Each model uses a different comparison group.
No coherent probability model. Predicted probabilities won’t sum to 1.
Inefficiency. Standard errors are larger than necessary.

A polytomous model fits all categories simultaneously in a single likelihood, sharing information across comparisons.

Setup and Data

library(tidyverse)
library(haven)
library(janitor)
library(broom)
library(knitr)
library(kableExtra)
library(gtsummary)
library(nnet)        # multinom() for polytomous regression
library(MASS)        # polr() for ordinal regression
library(brant)       # Brant test for proportional odds
library(ggeffects)

options(gtsummary.use_ftExtra = TRUE)
set_gtsummary_theme(theme_gtsummary_compact(set_theme = TRUE))

brfss_full <- read_xpt(
  "C:/Users/MY789914/OneDrive - University at Albany - SUNY/Desktop/Stat 553 (R)/In-Class R Lab Activities/LLCP2020.XPT"
) |>
  clean_names()

Building the Outcome: General Health

The BRFSS variable genhlth asks: “Would you say that in general your health is…”

1 = Excellent, 2 = Very Good, 3 = Good, 4 = Fair, 5 = Poor

We collapse this into a 3-level ordinal outcome to keep models tractable for class:

brfss_poly <- brfss_full |>
  mutate(
    # 3-level ordinal outcome
    genhlth_3 = case_when(
      genhlth %in% c(1, 2) ~ "Excellent/VG",
      genhlth == 3         ~ "Good",
      genhlth %in% c(4, 5) ~ "Fair/Poor",
      TRUE                 ~ NA_character_
    ),
    # Ordered factor for ordinal regression
    genhlth_ord = factor(genhlth_3,
      levels = c("Excellent/VG", "Good", "Fair/Poor"),
      ordered = TRUE),
    # Unordered factor for polytomous regression
    genhlth_nom = factor(genhlth_3,
      levels = c("Excellent/VG", "Good", "Fair/Poor")),
    # Predictors
    age = age80,
    sex = factor(sexvar, levels = c(1, 2), labels = c("Male", "Female")),
    bmi = ifelse(bmi5 > 0, bmi5 / 100, NA_real_),
    exercise = factor(case_when(
      exerany2 == 1 ~ "Yes",
      exerany2 == 2 ~ "No",
      TRUE          ~ NA_character_
    ), levels = c("No", "Yes")),
    income_cat = case_when(
      income2 %in% 1:8 ~ as.numeric(income2),
      TRUE             ~ NA_real_
    ),
    smoker = factor(case_when(
      smokday2 %in% c(1, 2) ~ "Current",
      smokday2 == 3          ~ "Former/Never",
      TRUE                   ~ NA_character_
    ), levels = c("Former/Never", "Current"))
  ) |>
  filter(
    !is.na(genhlth_ord), !is.na(age), age >= 18, !is.na(sex),
    !is.na(bmi), !is.na(exercise), !is.na(income_cat), !is.na(smoker)
  )

set.seed(1220)
brfss_poly <- brfss_poly |>
  dplyr::select(genhlth_ord, genhlth_nom, age, sex, bmi,
                exercise, income_cat, smoker) |>
  slice_sample(n = 5000)

saveRDS(brfss_poly,
  "C:/Users/MY789914/OneDrive - University at Albany - SUNY/Desktop/Stat 553 (R)/In-Class R Lab Activities/brfss_polytomous_2020.rds")

brfss_poly |>
  count(genhlth_ord) |>
  mutate(pct = round(100 * n / sum(n), 1)) |>
  kable(col.names = c("General Health", "N", "%"),
        caption = "Outcome Distribution") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Outcome Distribution
General Health	N	%
Excellent/VG	2380	47.6
Good	1624	32.5
Fair/Poor	996	19.9

Since genhlth_ord and genhlth_nom contain the same three levels (just coded as ordered vs. unordered factors), the distributions will be identical — but showing both makes the distinction between the two factor types explicit for the lecture, which is useful context for introducing polytomous vs. ordinal regression.

Part 1: Polytomous (Multinomial) Logistic Regression

The Model

For an outcome with \(K\) categories, we choose one as the reference (say, \(Y = 1\)) and fit \(K - 1\) logit equations:

\[ \log\left[\frac{\Pr(Y = k)}{\Pr(Y = 1)}\right] = \alpha_k + \beta_{k1} X_1 + \cdots + \beta_{kp} X_p, \quad k = 2, \ldots, K \]

Each comparison gets its own intercept and its own slopes. With 3 outcome categories and 5 predictors, that’s \(2 \times (1 + 5) = 12\) parameters.

The probabilities are recovered as:

\[ \Pr(Y = k \mid X) = \frac{\exp(\alpha_k + X^T \beta_k)}{1 + \sum_{j=2}^{K} \exp(\alpha_j + X^T \beta_j)} \]

By construction, the probabilities sum to 1.

Fitting in R: `nnet::multinom()`

mod_multi <- multinom(
  genhlth_nom ~ age + sex + bmi + exercise + income_cat + smoker,
  data = brfss_poly,
  trace = FALSE
)

summary(mod_multi)

## Call:
## multinom(formula = genhlth_nom ~ age + sex + bmi + exercise + 
##     income_cat + smoker, data = brfss_poly, trace = FALSE)
## 
## Coefficients:
##           (Intercept)        age  sexFemale        bmi exerciseYes income_cat
## Good        -1.317306 0.01497067 -0.1417121 0.05087197   -0.500636 -0.1693992
## Fair/Poor   -1.283880 0.02579060 -0.1257209 0.06741312   -1.338926 -0.3932631
##           smokerCurrent
## Good          0.4117899
## Fair/Poor     0.3436415
## 
## Std. Errors:
##           (Intercept)         age  sexFemale         bmi exerciseYes income_cat
## Good        0.2689762 0.002189706 0.06782701 0.005768280  0.08178912 0.01748213
## Fair/Poor   0.3296881 0.002916356 0.08583048 0.006778269  0.09155273 0.02090747
##           smokerCurrent
## Good         0.07723392
## Fair/Poor    0.09720041
## 
## Residual Deviance: 9287.557 
## AIC: 9315.557

Interpreting the Coefficients

The output gives two sets of coefficients, one for each non-reference category:

“Good” vs. “Excellent/VG”
“Fair/Poor” vs. “Excellent/VG”

Each \(\beta\) is interpreted as a log relative risk ratio (RRR). Exponentiating gives the relative risk ratio (sometimes called a relative odds ratio) for being in category \(k\) vs. the reference, per one-unit change in \(X\).

tidy(mod_multi, conf.int = TRUE, exponentiate = TRUE) |>
  dplyr::select(y.level, term, estimate, conf.low, conf.high, p.value) |>
  mutate(across(c(estimate, conf.low, conf.high), \(x) round(x, 3)),
         p.value = format.pval(p.value, digits = 3)) |>
  kable(col.names = c("Outcome", "Predictor", "RRR", "Lower", "Upper", "p"),
        caption = "Multinomial Logistic Regression: RRRs vs. Excellent/VG") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Multinomial Logistic Regression: RRRs vs. Excellent/VG
Outcome	Predictor	RRR	Lower	Upper	p
Good	(Intercept)	0.268	0.158	0.454	9.71e-07
Good	age	1.015	1.011	1.019	8.10e-12
Good	sexFemale	0.868	0.760	0.991	0.036679
Good	bmi	1.052	1.040	1.064	< 2e-16
Good	exerciseYes	0.606	0.516	0.712	9.30e-10
Good	income_cat	0.844	0.816	0.874	< 2e-16
Good	smokerCurrent	1.510	1.297	1.756	9.73e-08
Fair/Poor	(Intercept)	0.277	0.145	0.529	9.85e-05
Fair/Poor	age	1.026	1.020	1.032	< 2e-16
Fair/Poor	sexFemale	0.882	0.745	1.043	0.142987
Fair/Poor	bmi	1.070	1.056	1.084	< 2e-16
Fair/Poor	exerciseYes	0.262	0.219	0.314	< 2e-16
Fair/Poor	income_cat	0.675	0.648	0.703	< 2e-16
Fair/Poor	smokerCurrent	1.410	1.165	1.706	0.000407

Example interpretation. Holding all else constant, current smokers have a relative risk ratio of 1.41 (95% CI: 1.17–1.71) for reporting “Fair/Poor” health vs. “Excellent/VG” compared to former/never smokers. In other words, the relative risk of being in the “Fair/Poor” category (rather than “Excellent/VG”) is about 41% higher among current smokers.

Predicted Probabilities

ggpredict(mod_multi, terms = "income_cat [1:8]") |>
  plot() +
  labs(title = "Predicted Probability of Each Health Category by Income",
       x = "Income Category (1 = lowest, 8 = highest)",
       y = "Predicted Probability") +
  theme_minimal()

Interpretation. Holding all other predictors at their means, the predicted probability of reporting “Fair/Poor” health drops sharply across income levels — from 63% at the lowest income category to 17% at the highest. The predicted probability of “Excellent/VG” health shows the opposite pattern, rising from 11% to 49% across the same range. The probability of “Good” health remains comparatively stable (roughly 26%–35%), reflecting that income primarily shifts respondents between the two extremes rather than into or out of the “Good” category.

Part 2: Ordinal Logistic Regression

When categories are ordered, the polytomous model wastes information. It treats “Good” as just as different from “Excellent” as it is from “Fair/Poor”. An ordinal model exploits the ordering and produces fewer parameters and easier interpretation.

The Proportional Odds (Cumulative Logit) Model

The most common ordinal model is the proportional odds model, also called the cumulative logit model. Rather than modeling the probability of each category separately (as in the multinomial model), it models the probability of being at or below a given category. This respects the ordering of the outcome — “Excellent/VG” < “Good” < “Fair/Poor” in terms of health decline — and yields a single, consistent estimate of each predictor’s effect across the entire outcome scale.

The tradeoff for this parsimony is an additional assumption: the proportional odds assumption, which requires that the effect of each predictor is the same regardless of which cut-point is being modeled. This should be checked after fitting.

Define cumulative probabilities:

\[ \gamma_k = \Pr(Y \leq k \mid X) \]

The model is:

\[ \log\left[\frac{\Pr(Y \leq k)}{\Pr(Y > k)}\right] = \alpha_k - \beta_1 X_1 - \cdots - \beta_p X_p \]

Key features:

One intercept per cut-point (\(k = 1, \ldots, K - 1\))
A single set of slopes \(\beta\), shared across all cut-points
The “proportional odds” assumption: the effect of \(X\) on the log-odds of being at or below category \(k\) is the same for every \(k\)

With \(K = 3\) and 5 predictors, that’s \(2 + 5 = 7\) parameters (vs. 12 for multinomial).

In plain terms: imagine income predicts health. The proportional odds assumption says that the boost in odds of being in “Excellent/VG or better” (vs. “Good or worse”) from moving up one income level is the same as the boost in odds of being in “Excellent/VG or Good” (vs. “Fair/Poor”). The effect of income doesn’t change depending on which boundary you’re crossing — it’s a single, consistent shift up or down the health scale.

Fitting in R: `MASS::polr()`

mod_ord <- polr(
  genhlth_ord ~ age + sex + bmi + exercise + income_cat + smoker,
  data = brfss_poly,
  Hess = TRUE
)

summary(mod_ord)

## Call:
## polr(formula = genhlth_ord ~ age + sex + bmi + exercise + income_cat + 
##     smoker, data = brfss_poly, Hess = TRUE)
## 
## Coefficients:
##                  Value Std. Error t value
## age            0.01797   0.001867   9.627
## sexFemale     -0.12885   0.056835  -2.267
## bmi            0.04973   0.004534  10.967
## exerciseYes   -0.92167   0.064426 -14.306
## income_cat    -0.26967   0.014113 -19.107
## smokerCurrent  0.29313   0.064304   4.558
## 
## Intercepts:
##                   Value    Std. Error t value 
## Excellent/VG|Good   0.0985   0.2200     0.4478
## Good|Fair/Poor      1.8728   0.2212     8.4650
## 
## Residual Deviance: 9318.274 
## AIC: 9334.274

Odds Ratios and Interpretation

tidy(mod_ord, conf.int = TRUE, exponentiate = TRUE) |>
  filter(coef.type == "coefficient") |>
  dplyr::select(term, estimate, conf.low, conf.high) |>
  mutate(across(c(estimate, conf.low, conf.high), \(x) round(x, 3))) |>
  kable(col.names = c("Predictor", "OR", "Lower", "Upper"),
        caption = "Ordinal Logistic Regression: Cumulative ORs") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Ordinal Logistic Regression: Cumulative ORs
Predictor	OR	Lower	Upper
age	1.018	1.014	1.022
sexFemale	0.879	0.786	0.983
bmi	1.051	1.042	1.060
exerciseYes	0.398	0.351	0.451
income_cat	0.764	0.743	0.785
smokerCurrent	1.341	1.182	1.521

Interpretation. An OR of \(e^{\beta}\) for a predictor means: a one-unit increase in \(X\) multiplies the odds of being in a worse health category (i.e., \(Y > k\) vs. \(Y \leq k\)) by \(e^{\beta}\), and this is true at every cut-point. Here, \(\beta\) is the estimated coefficient from the model (the log-odds), and \(e^{\beta}\) is simply that coefficient exponentiated — converting it from the log-odds scale to an odds ratio that is easier to interpret.

For example, if smoking has OR = 1.6, then current smokers have 1.6 times the odds of reporting Good or worse health (vs. Excellent/VG), AND 1.6 times the odds of reporting Fair/Poor (vs. Excellent/VG or Good).

Predicted Probabilities

ggpredict(mod_ord, terms = "income_cat [1:8]") |>
  plot() +
  labs(title = "Ordinal Model: Predicted Probability of Each Health Category",
       x = "Income Category", y = "Predicted Probability") +
  theme_minimal()

Part 3: Checking the Proportional Odds Assumption

The proportional odds assumption is strong. It states that the effect of each predictor on the log-odds of being at or below any cut-point is constant across all cut-points — in other words, one \(\beta\) summarizes the predictor’s effect across the entire outcome scale. If this holds, the ordinal model is both valid and parsimonious. If it fails, that single \(\beta\) is a misleading average of effects that actually differ across thresholds, and a more flexible model is needed.

There are two complementary ways to evaluate this assumption: a visual check and a formal test.

Visual Check: Empirical Logits

The idea is to fit separate binary logistic regressions at each cumulative cut-point and compare the coefficients. If the proportional odds assumption holds, the log-odds coefficients should be roughly the same at both cut-points — the two sets of points should overlap or sit close together. Large separation between cut-points for a given predictor is a warning sign.

cut1 <- glm(I(as.numeric(genhlth_ord) <= 1) ~ age + sex + bmi + exercise + income_cat + smoker,
            data = brfss_poly, family = binomial)
cut2 <- glm(I(as.numeric(genhlth_ord) <= 2) ~ age + sex + bmi + exercise + income_cat + smoker,
            data = brfss_poly, family = binomial)

bind_rows(
  broom::tidy(cut1, conf.int = TRUE) |> mutate(cutpoint = "\u2264 Excellent/VG"),
  broom::tidy(cut2, conf.int = TRUE) |> mutate(cutpoint = "\u2264 Good")
) |>
  filter(term != "(Intercept)") |>
  ggplot(aes(x = estimate, y = term, color = cutpoint, shape = cutpoint)) +
  geom_vline(xintercept = 0, linetype = "dashed", color = "gray50") +
  geom_pointrange(aes(xmin = conf.low, xmax = conf.high),
                  position = position_dodge(width = 0.5)) +
  labs(
    title = "Visual Check: Proportional Odds Assumption",
    subtitle = "Coefficients should overlap across cut-points if assumption holds",
    x = "Log-Odds Coefficient",
    y = "Predictor",
    color = "Cut-point",
    shape = "Cut-point"
  ) +
  theme_minimal()

Predictors whose point estimates are far apart between cut-points (e.g., smokerCurrent, sexFemale) are candidates for violating the assumption — confirm with the Brant test below.

Formal Test: Brant Test

Omnibus p > 0.05: assumption holds for the model overall
Predictor-level p < 0.05: assumption fails for that variable

brant_out <- brant(mod_ord)

## -------------------------------------------- 
## Test for X2  df  probability 
## -------------------------------------------- 
## Omnibus      35.87   6   0
## age      0.06    1   0.81
## sexFemale    0.89    1   0.34
## bmi      7.1 1   0.01
## exerciseYes  10.58   1   0
## income_cat   10.54   1   0
## smokerCurrent    7.76    1   0.01
## -------------------------------------------- 
## 
## H0: Parallel Regression Assumption holds

# as.data.frame(brant_out) |>
#   tibble::rownames_to_column("Predictor") |>
#   mutate(
#     X2          = round(X2, 2),
#     probability = ifelse(probability < 0.001, "< 0.001", sprintf("%.3f", probability)),
#     Fails       = ifelse(Predictor == "Omnibus",
#                          ifelse(as.numeric(gsub("< ", "", probability)) < 0.05 | probability == "< 0.001", "Yes (overall)", "No (overall)"),
#                          ifelse(probability == "< 0.001" | as.numeric(gsub("< ", "", probability)) < 0.05, "⚠ Yes", " "))
#   ) |>
#   kable(
#     col.names = c("Predictor", "X²", "df", "p-value", "Assumption Violated?"),
#     caption   = "Brant Test for Proportional Odds Assumption",
#     align     = c("l", "r", "r", "r", "c")
#   ) |>
#   kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) |>
#   row_spec(0, bold = TRUE) |>
#   row_spec(1, bold = TRUE, background = "#f0f0f0")

What If It Fails?

Use the multinomial model instead (loses parsimony, gains flexibility)
Use a partial proportional odds model (some \(\beta\)s vary across cut-points)
Collapse outcome categories
Use a different link (continuation-ratio, adjacent-category)

Part 4: Comparing the Two Models

tibble(
  Feature = c("Outcome type", "Number of equations", "Parameters (3 cats, 5 preds)",
              "Key assumption", "Best when"),
  `Multinomial (multinom)` = c("Nominal or ordinal",
                                "K - 1 = 2",
                                "12",
                                "None beyond independence",
                                "Categories unordered or PO fails"),
  `Ordinal (polr)`         = c("Ordinal only",
                                "1 (with K - 1 cut-points)",
                                "7",
                                "Proportional odds",
                                "Categories ordered AND PO holds")
) |>
  kable(caption = "Multinomial vs. Ordinal Logistic Regression") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Multinomial vs. Ordinal Logistic Regression
Feature	Multinomial (multinom)	Ordinal (polr)
Outcome type	Nominal or ordinal	Ordinal only
Number of equations	K - 1 = 2	1 (with K - 1 cut-points)
Parameters (3 cats, 5 preds)	12	7
Key assumption	None beyond independence	Proportional odds
Best when	Categories unordered or PO fails	Categories ordered AND PO holds

Likelihood Comparison

Both models are fit by maximum likelihood. We can compare their fits with AIC (lower = better):

tibble(
  Model = c("Multinomial", "Ordinal (proportional odds)"),
  AIC   = c(AIC(mod_multi), AIC(mod_ord)),
  df    = c(mod_multi$edf, length(coef(mod_ord)) + length(mod_ord$zeta))
) |>
  mutate(AIC = round(AIC, 1)) |>
  kable(caption = "Model Comparison") |>
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Model Comparison
Model	AIC	df
Multinomial	9315.6	14
Ordinal (proportional odds)	9334.3	8

Tradeoff: The ordinal model is more parsimonious. If proportional odds holds, it has lower AIC and tighter confidence intervals. If PO fails, the multinomial model is more honest.

Summary

Concept	Key Point
Polytomous outcome	Use `nnet::multinom()` for \(K > 2\) categories
Coefficients	\(K - 1\) sets, each interpreted as log relative risk ratios vs. reference
Ordinal outcome	Use `MASS::polr()` to exploit ordering
Cumulative logit	Single set of \(\beta\)s describes all cut-points
Proportional odds	Strong assumption; check with `brant()`
Model choice	Ordinal if ordered AND PO holds; otherwise multinomial

Next Lecture (April 21)

Logistic regression applications and case studies
Putting it all together: model building strategy
Reporting results for publication

References

Kleinbaum, D. G., Kupper, L. L., Nizam, A., & Rosenberg, E. S. (2013). Applied Regression Analysis and Other Multivariable Methods (5th ed.), Chapter 23.
Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied Logistic Regression (3rd ed.), Chapter 8.
Agresti, A. (2010). Analysis of Ordinal Categorical Data (2nd ed.). Wiley.

Part 5: In-Class Lab Activity

EPI 553 — Polytomous and Ordinal Regression Lab Due: End of class, April 16, 2026

Instructions

Complete the four tasks below using brfss_polytomous_2020.rds. Submit a knitted HTML file via Brightspace.

Data

Variable	Description	Type
`genhlth_ord`	General health (Excellent/VG < Good < Fair/Poor)	Ordered factor (outcome)
`genhlth_nom`	General health, unordered	Factor (outcome)
`age`	Age in years	Numeric
`sex`	Male / Female	Factor
`bmi`	Body mass index	Numeric
`exercise`	Exercised in past 30 days (No/Yes)	Factor
`income_cat`	Household income (1-8)	Numeric
`smoker`	Former/Never vs. Current	Factor

library(tidyverse)
library(broom)
library(knitr)
library(kableExtra)
library(gtsummary)
library(nnet)
library(MASS)
library(brant)
library(ggeffects)

options(gtsummary.use_ftExtra = TRUE)
set_gtsummary_theme(theme_gtsummary_compact(set_theme = TRUE))

brfss_poly <- readRDS("brfss_polytomous_2020.rds") %>%
  dplyr::rename(age_years = age)

Task 1: Explore the Outcome (15 points)

1a. (5 pts) Create a frequency table of genhlth_ord showing N and percentage for each category.

# Frequency table with counts and percentages
genhlth_table <- brfss_poly %>%
  count(genhlth_ord) %>%
  mutate(percent = 100 * n / sum(n))

genhlth_table %>%
  kable(digits = 1, col.names = c("General Health", "N", "Percent (%)")) %>%
  kable_styling(full_width = FALSE)

General Health	N	Percent (%)
Excellent/VG	2380	47.6
Good	1624	32.5
Fair/Poor	996	19.9

1b. (5 pts) Create a stacked bar chart showing the distribution of general health by smoker status.

ggplot(brfss_poly, aes(x = smoker, fill = genhlth_ord)) +
  geom_bar(position = "fill") +
  labs(
    title = "Distribution of General Health by Smoking Status",
    x = "Smoking Status",
    y = "Proportion",
    fill = "General Health"
  ) +
  scale_y_continuous(labels = scales::percent) +
  theme_minimal()

1c. (5 pts) Use tbl_summary() to create a descriptive table stratified by genhlth_ord with at least 4 predictors.

table1 <- brfss_poly %>%
  dplyr::select(genhlth_ord, age_years, sex, bmi, exercise, income_cat, smoker) %>%
  tbl_summary(
    by = genhlth_ord,
    label = list(
      age_years ~ "Age (years)",
      sex ~ "Sex",
      bmi ~ "BMI",
      exercise ~ "Exercised in past 30 days",
      income_cat ~ "Household income category",
      smoker ~ "Smoking status"
    ),
    statistic = list(
      all_continuous() ~ "{mean} ({sd})",
      all_categorical() ~ "{n} ({p}%)"
    ),
    digits = all_continuous() ~ 1
  ) %>%
  bold_labels() %>%
  modify_caption("**Table 1. Descriptive Characteristics by General Health Status, BRFSS 2020**")

table1

**Table 1. Descriptive Characteristics by General Health Status, BRFSS 2020**
Characteristic	Excellent/VG N = 2,380¹	Good N = 1,624¹	Fair/Poor N = 996¹
Age (years)	54.7 (16.5)	57.9 (16.3)	60.8 (14.6)
Sex
Male	1,315 (55%)	906 (56%)	496 (50%)
Female	1,065 (45%)	718 (44%)	500 (50%)
BMI	27.4 (5.2)	29.1 (6.4)	30.0 (8.0)
Exercised in past 30 days	1,999 (84%)	1,161 (71%)	470 (47%)
Household income category
1	39 (1.6%)	72 (4.4%)	114 (11%)
2	69 (2.9%)	92 (5.7%)	110 (11%)
3	124 (5.2%)	128 (7.9%)	151 (15%)
4	174 (7.3%)	188 (12%)	150 (15%)
5	198 (8.3%)	197 (12%)	124 (12%)
6	354 (15%)	249 (15%)	111 (11%)
7	443 (19%)	289 (18%)	112 (11%)
8	979 (41%)	409 (25%)	124 (12%)
Smoking status
Former/Never	1,692 (71%)	1,015 (63%)	597 (60%)
Current	688 (29%)	609 (38%)	399 (40%)
¹ Mean (SD); n (%)

Interpretation: Table 1 shows that individuals reporting Fair/Poor health were older, had higher BMI, were less likely to exercise, and were more likely to be current smokers compared to those with Excellent/Very Good health. They were also more concentrated in lower income categories, while higher income was more common among those with better health. The Good health group generally fell between these extremes. Overall, these patterns suggest that demographic and behavioral factors are associated with worse self-rated health.

Task 2: Multinomial Logistic Regression (20 points)

2a. (5 pts) Fit a multinomial logistic regression model predicting genhlth_nom from at least 4 predictors using multinom().

multi_model <- multinom(
  genhlth_nom ~ age_years + sex + bmi + exercise + income_cat + smoker,
  data = brfss_poly
)

## # weights:  24 (14 variable)
## initial  value 5493.061443 
## iter  10 value 4939.569740
## iter  20 value 4643.779154
## final  value 4643.778340 
## converged

2b. (10 pts) Report the relative risk ratios (exponentiated coefficients) with 95% CIs in a clean table.

multi_results <- broom::tidy(multi_model, conf.int = TRUE, exponentiate = TRUE)

multi_results %>%
  dplyr::select(y.level, term, estimate, conf.low, conf.high) %>%
  dplyr::rename(
    Outcome = y.level,
    Predictor = term,
    RRR = estimate,
    CI_Lower = conf.low,
    CI_Upper = conf.high
  ) %>%
  knitr::kable(digits = 2, caption = "Table 2. Multinomial Logistic Regression Results (RRR, 95% CI)")

Table 2. Multinomial Logistic Regression Results (RRR, 95% CI)
Outcome	Predictor	RRR	CI_Lower	CI_Upper
Good	(Intercept)	0.27	0.16	0.45
Good	age_years	1.02	1.01	1.02
Good	sexFemale	0.87	0.76	0.99
Good	bmi	1.05	1.04	1.06
Good	exerciseYes	0.61	0.52	0.71
Good	income_cat	0.84	0.82	0.87
Good	smokerCurrent	1.51	1.30	1.76
Fair/Poor	(Intercept)	0.28	0.15	0.53
Fair/Poor	age_years	1.03	1.02	1.03
Fair/Poor	sexFemale	0.88	0.75	1.04
Fair/Poor	bmi	1.07	1.06	1.08
Fair/Poor	exerciseYes	0.26	0.22	0.31
Fair/Poor	income_cat	0.67	0.65	0.70
Fair/Poor	smokerCurrent	1.41	1.17	1.71

2c. (5 pts) Interpret the RRR for one predictor in the “Fair/Poor vs. Excellent/VG” comparison.

For the Fair/Poor vs. Excellent/Very Good comparison, the relative risk ratio for exercise is 0.26 (95% CI: 0.22–0.31). This indicates that individuals who exercised in the past 30 days have approximately 74% lower relative risk of reporting Fair/Poor health rather than Excellent/Very Good health, compared to those who did not exercise, holding all other variables constant. This suggests that physical activity is strongly associated with better self-rated health.

Task 3: Ordinal Logistic Regression (25 points)

3a. (5 pts) Fit an ordinal logistic regression model with the same predictors using polr().

ord_model <- polr(
  genhlth_ord ~ age_years + sex + bmi + exercise + income_cat + smoker,
  data = brfss_poly,
  Hess = TRUE
)
summary(ord_model)

## Call:
## polr(formula = genhlth_ord ~ age_years + sex + bmi + exercise + 
##     income_cat + smoker, data = brfss_poly, Hess = TRUE)
## 
## Coefficients:
##                  Value Std. Error t value
## age_years      0.01797   0.001867   9.627
## sexFemale     -0.12885   0.056835  -2.267
## bmi            0.04973   0.004534  10.967
## exerciseYes   -0.92167   0.064426 -14.306
## income_cat    -0.26967   0.014113 -19.107
## smokerCurrent  0.29313   0.064304   4.558
## 
## Intercepts:
##                   Value    Std. Error t value 
## Excellent/VG|Good   0.0985   0.2200     0.4478
## Good|Fair/Poor      1.8728   0.2212     8.4650
## 
## Residual Deviance: 9318.274 
## AIC: 9334.274

3b. (5 pts) Report the cumulative ORs with 95% CIs.

ord_results <- tidy(ord_model, conf.int = TRUE, exponentiate = TRUE)

ord_results %>%
  dplyr::filter(!grepl("\\|", term)) %>%
  dplyr::mutate(
    term = dplyr::recode(term,
      "age_years" = "Age (years)",
      "sexFemale" = "Female (vs Male)",
      "bmi" = "BMI",
      "exerciseYes" = "Exercise (Yes vs No)",
      "income_cat" = "Household income category",
      "smokerCurrent" = "Current smoker (vs Former/Never)"
    )
  ) %>%
  dplyr::select(term, estimate, conf.low, conf.high) %>%
  dplyr::rename(
    Predictor = term,
    OR = estimate,
    CI_Lower = conf.low,
    CI_Upper = conf.high
  ) %>%
  knitr::kable(digits = 2, caption = "Table 3. Ordinal Logistic Regression Results (Cumulative OR, 95% CI)")

Table 3. Ordinal Logistic Regression Results (Cumulative OR, 95% CI)
Predictor	OR	CI_Lower	CI_Upper
Age (years)	1.02	1.01	1.02
Female (vs Male)	0.88	0.79	0.98
BMI	1.05	1.04	1.06
Exercise (Yes vs No)	0.40	0.35	0.45
Household income category	0.76	0.74	0.78
Current smoker (vs Former/Never)	1.34	1.18	1.52

3c. (5 pts) Interpret one OR in plain language, making sure to mention the “at every cut-point” property.

The cumulative odds ratio for exercise is 0.40 (95% CI: 0.35–0.45), indicating that individuals who exercised in the past 30 days have 60% lower odds of being in a worse general health category compared to those who did not exercise, holding all other variables constant. This relationship holds at every cut-point of the outcome, meaning that individuals who exercised have lower odds of being in Good or Fair/Poor vs. Excellent/Very Good, as well as Fair/Poor vs. Excellent/Very Good/Good. This suggests that physical activity is consistently associated with better self-rated health across all levels of general health.

3d. (10 pts) Use ggpredict() to plot predicted probabilities of each health category across a continuous predictor of your choice.

# Ensure outcome is ordered factor
brfss_poly$genhlth_ord <- factor(
  brfss_poly$genhlth_ord,
  levels = c("Excellent/VG", "Good", "Fair/Poor"),
  ordered = TRUE
)

# Fit ordinal model
ord_model <- MASS::polr(
  genhlth_ord ~ age_years + sex + bmi + exercise + income_cat + smoker,
  data = brfss_poly,
  Hess = TRUE
)

# Generate predicted probabilities across BMI
pred <- ggpredict(ord_model, terms = "bmi [all]")

# Plot predicted probabilities
plot(pred)

Interpretation: The predicted probabilities indicate that as BMI increases, the probability of reporting Excellent/Very Good health decreases, while the probability of Fair/Poor health increases. The probability of Good health peaks at moderate BMI levels and then declines at higher BMI. Overall, these results show a clear shift toward worse self-rated health as BMI increases.

Task 4: Check Assumptions and Compare (15 points)

4a. (5 pts) Run the Brant test for proportional odds. Does the assumption hold?

library(brant)

brant(ord_model)

## -------------------------------------------- 
## Test for X2  df  probability 
## -------------------------------------------- 
## Omnibus      35.87   6   0
## age_years    0.06    1   0.81
## sexFemale    0.89    1   0.34
## bmi      7.1 1   0.01
## exerciseYes  10.58   1   0
## income_cat   10.54   1   0
## smokerCurrent    7.76    1   0.01
## -------------------------------------------- 
## 
## H0: Parallel Regression Assumption holds

Interpretation: The Brant test was statistically significant (Omnibus χ² = 35.87, p < 0.001), indicating that the proportional odds assumption is violated. This suggests that the relationship between predictors and general health is not consistent across outcome cut-points. In particular, variables such as BMI, exercise, income, and smoking show evidence of violating the assumption, meaning their effects differ across levels of general health.

4b. (5 pts) Compare the AIC of the multinomial and ordinal models. Which fits better?

AIC(multi_model)

## [1] 9315.557

AIC(ord_model)

## [1] 9334.274

Interpretation: The AIC for the multinomial model is 9315.56, and the AIC for the ordinal model is 9334.27. Since the multinomial model has the lower AIC, it provides a better fit to the data.

4c. (5 pts) Based on your results, which model would you recommend reporting? Justify in 2-3 sentences.

The multinomial logistic regression model is preferred because the proportional odds assumption for the ordinal model is violated. In addition, the multinomial model has a lower AIC, indicating better model fit. Therefore, the multinomial model provides a more appropriate and flexible representation of the relationships between predictors and general health.

Completion credit (25 points): Awarded for a complete, good-faith attempt. Total: 75 + 25 = 100 points.

End of Lab Activity

Polytomous and Ordinal Logistic Regression

EPI 553 — Principles of Statistical Inference II

Maral Yerdenova

April 16, 2026

Introduction

Why Not Just Run Multiple Binary Models?

Setup and Data

Building the Outcome: General Health

Part 1: Polytomous (Multinomial) Logistic Regression

The Model

Fitting in R: `nnet::multinom()`

Interpreting the Coefficients

Predicted Probabilities

Part 2: Ordinal Logistic Regression

The Proportional Odds (Cumulative Logit) Model

Fitting in R: `MASS::polr()`

Odds Ratios and Interpretation

Predicted Probabilities

Part 3: Checking the Proportional Odds Assumption

Visual Check: Empirical Logits

Formal Test: Brant Test

What If It Fails?

Part 4: Comparing the Two Models

Likelihood Comparison

Summary

Next Lecture (April 21)

References

Part 5: In-Class Lab Activity

Instructions

Data

Task 1: Explore the Outcome (15 points)

Task 2: Multinomial Logistic Regression (20 points)

Task 3: Ordinal Logistic Regression (25 points)

Task 4: Check Assumptions and Compare (15 points)

Polytomous and Ordinal Logistic Regression

EPI 553 — Principles of Statistical Inference II

Maral Yerdenova

April 16, 2026

Introduction

Why Not Just Run Multiple Binary Models?

Setup and Data

Building the Outcome: General Health

Part 1: Polytomous (Multinomial) Logistic Regression

The Model

Fitting in R: nnet::multinom()

Interpreting the Coefficients

Predicted Probabilities

Part 2: Ordinal Logistic Regression

The Proportional Odds (Cumulative Logit) Model

Fitting in R: MASS::polr()

Odds Ratios and Interpretation

Predicted Probabilities

Part 3: Checking the Proportional Odds Assumption

Visual Check: Empirical Logits

Formal Test: Brant Test

What If It Fails?

Part 4: Comparing the Two Models

Likelihood Comparison

Summary

Next Lecture (April 21)

References

Part 5: In-Class Lab Activity

Instructions

Data

Task 1: Explore the Outcome (15 points)

Task 2: Multinomial Logistic Regression (20 points)

Task 3: Ordinal Logistic Regression (25 points)

Task 4: Check Assumptions and Compare (15 points)

Fitting in R: `nnet::multinom()`

Fitting in R: `MASS::polr()`